Determinants, part II
Math 130 Linear Algebra
D Joyce, Fall 2015
So far we’ve only defined determinants of 2 × 2 and 3 × 3 matrices. The 2 × 2 determinants
had 2 terms, while the × determinants had 6 terms.
There are many ways that general n × n determinants can be defined. We’ll first define
a determinant function in terms of characterizing properties that we want it to have. Then
we’ll use the construction of a determinant following the method given in the text.
Definition 1. A determinant function assigns to each square matrix A a scalar associated
to the matrix, denoted det(A) or |A| such that
(1). The determinant of an n × n identity matrix I is 1. |I| = 1.
(2). If the matrix B is identical to the matrix A except the entries in one of the rows
of B are each equal to the corresponding entries of A multiplied by the same scalar c, then
|B| = c |A|.
(3). If the matrices A, B, and C are identical except for the entries in one row, and
for that row an entry in A is found by adding the corresponding entries in B and C, then
|A| = |B| + |C|.
(4). If the matrix B is the result of exchanging two rows of A, then the determinant of B
is the negation of the determinant of A.
The second and third conditions together say that determinants are linear in each row.
Usually that’s phrased by saying determinants are mulitlinear in their rows.
The last condition says it’s an “alternating” function.
These conditions are enough to characterize the determinant, but they don’t show such
a determinant function exists and is unique. We’ll show both existence and uniqueness, but
start with uniqueness. First, we’ll note a couple of properties that determinant functions
have that follow from the definition.
Theorem 2. A determinant function has the following four properties.
(a). The determinant of any matrix with an entire row of 0’s is 0.
(b). The determinant of any matrix with two identical rows is 0.
(c). If one row of a matrix is a multiple of another row, then its determinant is 0.
(d). If a multiple of one row of a matrix is added to another row, then the resulting matrix
has the same determinant as the original matrix.
Proof. Property (a) follows from the second statement in the definition. If A has a whole row
of 0’s, then using that row and c = 0 in the second statement of the definition, then B = A,
so |A| = 0|A|. Therefore, |A| = 0.
Property (b) follows from the fourth statement in the definition. If you exchange the two
identical rows, the result is the original matrix, but its determinant is negated. The only
scalar which is its own negation is 0. Therefore, the determinant of the matrix is 0.
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Property (c) follows from the second statement in the definition and property (b).
Property (d) follows from the third statement in the definition and property (c). q.e.d.
Now we can show the uniqueness of determinant functions.
Theorem 3. There is at most one determinant function.
Proof. The four properties that determinants are enough to find the value of the determinant
of a matrix.
Suppose a matrix A has more than one nonzero entry in a row. Then using the sum
property of multilinearity, we know
|A| = |A1 | + |A2 | + · · · + |An |
where Aj is the matrix that looks just like A except in that row, all the entries are 0 except
the j th one which is the j th entry of that row in A.
That means we can reduce the question of evaluating determinants of general matrices to
evaluating determinants of matrices that have at most one nonzero entry in each row.
By property (a) in the preceding theorem, if the matrix has a row of all 0’s, its determinant
is 0. Thus, we only need to consider matrices that have exactly one nonzero entry in each
row.
Using the scalar property of multilinearity, we can further assume that the nonzero entry
in that row is 1.
Now we’re down to evaluating determinants that only have entries of 0’s and 1’s with
exactly one 1 in each row.
If two of those rows have the 1’s in the same column, then by property (b) of the preceding
theorem, that matrix has determinant 0.
Now the only matrices left to consider are those that only have entries of 0’s and 1’s with
exactly one 1 in each row and column. These are called permutation matrices.
Using alternation, the fourth condition, the rows can be interchanged until the 1’s only
lie on the main diagonal (each interchange negating the determinant).
Finally, we’re left with the identity matrix, but by the first condition in the definition, its
determinant is 0.
Thus, the value of the determinant of of every matrix is determined by the definition.
There can be only one determinant function.
q.e.d.
Although that argument shows that there’s at most one function with those four properties, it doesn’t show that there is such a function. In other words, we haven’t shown the
properties are consistent. We need some way to construct a function with those properties,
and we’ll do that with a “cofactor construction”.
In the process of proving that theorem, we found out what the determinant of a permutation matrix had to be. Let’s state that as a corollary since it’s an important result in its
own right.
Corollary 4. A permutation matrix is a square matrix that only has 0’s and 1’s as its entries
with exactly one 1 in each row and column.
The determinant of a permutation matrix will have to be either 1 or −1 depending on
whether it takes an even number or an odd number of row interchanges to convert it to the
identity matrix.
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The cofactor construction. Although there are many ways to construct the determinant
function, here’s one that uses the cofactors.
Example 5. Probably the best way to understand cofactor expansion is to use it in an
example. In this example, we’ll do cofactor expansion across the first row of a matrix. Let’s
start with a typical 3 × 3 matrix


5
2 −1
3 
A =  4 −2
6
1
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For each of the three entries across the first row, we’ll cross out the row and column of that
entry to get a 2 × 2 matrix called a cofactor matrix. Then we’ll multiply each entry in the
first row by the determinant of its cofactor matrix, and then add or subtract the resulting
products to get the determinant.
−2 3 4 −2 4 3 −2·
|A| = 5 · 6 4 + (−1) · 6
1 1 4 Thus, we’ve reduced the problem of finding one 3 × 3 determinant to finding three 2 × 2
determinants. The rest of the computation goes like this:
|A| = 5 · (−11) − 2 · (−2) + (−1) · 16 = −55 + 4 − 16 = −67
Now, notice how we alternately added and subtracted the three terms to get the product.
When you expand across the first row, the signs you use are +−+. You could instead expand
across the second row, you’ll use the signs − + −. The sign you use for the ij th entry is
(−1)i+j . This corresponds to this checkerboard pattern for the entries of the matrix


+ − +
 − + − 
+ − +
which starts with + in the upper left corner.
We’ll construct a determinant function using expansion across the first row of the matrix.
We’ll call this function we’re constructing a determinant and use the standard notation for
it, but we have to show that it has the four properties required for a determinant function.
This cofactor construction is an recursive definition (also called an inductive definition)
since the determinant of an n×n matrix is defined in terms of determinants of (n−1)×(n−1)
matrices. Like all recursive definitions, we need a base case, and that will be when n = 1.
We’ll define the determinant of a 1 × 1 matrix to be its entry.
We need to define cofactor matrices before we can get to the construction.
Definition 6. If A is an n × n matrix, i is a row number, and j is a column number, then
the ij th cofactor matrix, denoted Ãij , is the matrix that results from deleting the ith row and
the j th column from A. The cofactor matrix Ãij is an (n − 1) × (n − 1) matrix.
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Our recursive construction for determinants “expands” the matrix along the first row. We
define |A| as an alternating sum of the determinants of its cofactor matrices
|A| =
n
X
(−1)1+j A1j |Ã1j | = A11 |Ã11 | − A12 |Ã12 | + A13 |Ã13 | − · · · ± A1n |Ã1n |
j=1
This definition gives the same values for 2 × 2 and 3 × 3 determinants we saw before.
The following theorem is formally proven in steps in the text.
Theorem 7. The four characterizing properties of determinants listed above are satisfied by
the cofactor definition of determinants.
The real purpose of this theorem is just to show the existence of a determinant function.
We usually won’t use the construction to actually compute determinants because there are
much better ways. There are some special matrices where this construction gives a quick
method of evaluation, namely diagonal, upper triangular, and lower triangular matrices.
Theorem 8. The determinant of a diagonal matrix or a triangular matrix is the product of
its diagonal entries.
Proof. In each of these three kinds of matrices, if you use cofactor expansion along the first
row, the only term in the expansion that’s not zero is the first, and that term is the product
of the first entry and its cofactor matrix, which by induction will have a determinant which
is the product of its diagonal entries.
q.e.d.
Math 130 Home Page at http://math.clarku.edu/~ma130/
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